Macromolecules 1985, 18, 1157-1162
friction coefficient increases more rapidly with concentration than is expected from the nondraining sphere of the blob for some still unclarified reason, such as the decrease of the free volume.1° If this is the case, the blobscaling result, eq 18, may hold valid at still lower concentrations or in some other (less viscous) solvent. An experiment at lower concentrations with a polymer of M = 2.5 X lo7 failed due to the degradation of the polymer. Measurements in less viscous solvents and higher frequencies will be valuable in elucidating the content of the parameter 7,. Measurements at higher frequencies and lower concentrations may be valuable also for detecting the onset of entanglement in the semidilute solutions.
Acknowledgment. We acknowledge support from a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, Japan (No. 59550612). References and Notes (1) . . Ferrv. J. D. “Viscoelastic ProDerties of Polvmers”. 3rd ed.: Wiley: New York, 1980; Chapter 10. (2) Doi, M. J. Polym. Sci., Polym. Phys. Ed. 1980, 18,1005.
1157
(3) de Gennes, P.-G. “Scaling Concepts in Polymer Physics”;
Cornel1 University Press: London, 1979. (4) See: Tirrell, M. Rubber Chem. Technol. 1984, 57,523. (5) Adam, M.; Delsanti, M. J. Phys. (Paris) 1983, 44,1185. (6) Osaki, K.; Nishizawa, K.; Kurata, M. Macromolecules 1982,15, 1068. (7) Osaki. K.: Nishizawa., K.:, Kurata. M. Nihon Reoroii Gakkaishi 1982, ’10,’169. (8) Brueaaeman. B. G.; Minnick, M. G.; Schrae, - J. L. Macromolecule;-1978, 11, 119. (9) Tamura, M.; Kurata, M.; Osaki, K.; Einaga, Y.; Kimura, S. Bull. Inst. Chem. Res., Kyoto Uniu. 1971, 49, 43. (10) See ref 1, Chapter 11. (11) See ref 1, Chapter 4. (12) Raju, V. R.; Menezes, E. V.; Marin, G.; Graessley, W. W. Macromolecules 1981, 14,1668. 1953. 21. 1272. (13) Rouse. P. E. J. Chem. Phvs. ” (14) See ref 1, Appendix E. (15) . . Osaki, K.: Kimura. S.: Kurata. M. J.Polvm. Sci.. Polvm. Phvs. Ed. 1981; 19, 517.’ (16) Kimura, S. Thesis (Dr. of Engineering), Kyoto University, 1982. (17) Doi, M.; Edwards, S. F. J. Chem. SOC.,Faraday Trans. 2 1978, 74,1789. (18) Onogi, S.; Masuda, T.; Kitagawa, K. Macromolecules 1970,3, 109. ,
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Viscoelasticity of Randomly Cross-Linked Polymer Networks. Relaxation of Dangling Chains? John G. Curro* Sandia National Laboratories, Albuquerque, New Mexico 87185
Dale S. Pearson Exxon Research and Engineering Company, Corporate Research Science Laboratories, Annandale, New Jersey 08801
Eugene Helfand AT&T Bell Laboratories, Murray Hill, New Jersey 07974. Received October 9, 1984 ABSTRACT A model baaed on the diffusion of dangling chains (in the presence of topological constraints) is used to predict the long-time, viscoelastic relaxation of randomly cross-linked polymer networks. In this calculation, the Pearson-Helfand theory for polymer stars is modified to account for the random distribution of chain lengths present in networks. An exact, numerical result is obtained which is found to agree with the previous Curro-Pincus theory at long times. In this limit, the theoretical relaxation modulus can be approximated by a power-law time dependence as in the phenomenological Thirion-Chasset equation. The cross-link density dependence of the exponent is in agreement with experimental data on natural rubber.
network imperfections consisting of dangling chains ends.I Introduction Ferry1 has postulated that the relaxation of these dangling Polymer networks typically exhibit exceedingly long ends, in the presence of entanglements, is responsible for viscoelastic relaxation times.l For example, it can take the observed long relaxation times. The Curro-Pincus hundreds of hours for lightly cross-linked natural rubber to equilibrate in a stress relaxation or creep e ~ p e r i m e n t . ~ ? ~ theory, as well as the present theory, employs a model based on the diffusion of dangling chains in the presence Recently Curro and Pincus4 developed a theory for this of topological constraints. long-time viscoelastic behavior of networks which is in The diffusion of a linear polymer chain is strongly agreement with experimental observations. The Currosuppressed in a melt because of the presence of entanPincus theory is based on the ideas of de Genness regarding glements which act as topological constraints. The diffuthe reptation of branched polymer molecules in the sion mechanism in this case is thought to consist of a presence of topological constraints. The purpose of the reptation or “snakelike” motion of the chain.* The topopresent investigation is to develop a more quantitative logical nature of a dangling chain or polymer star molecule, theory by extending the recent results of Pearson and however, prevents the ordinary reptation process from Helfand6 on star molecules to randomly cross-linked occurring. De Gennes5 has postulated that such a polymer networks. branched molecule relaxes to its equilibrium state by the A polymer network randomly cross-linked from polymer branched chain retracing or retracting along its primitive chains of finite molecular weight invariably contains path. In the recent Pearson-Helfand t h e o g for polymer stars, the path retraction process is modeled by considering the This paper is dedicated to Dr. Pierre Thirion on the occasion of his retirement. chain end to be undergoing Brownian mction in a suitable 0024-9297/85/2218-1157$01.50/0
0 1985 American Chemical Society
Macromolecules, Vol. 18, No. 6, 1985
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potential field. The probability distribution for branch retraction then is the solution of the Smoluchowski equationg for the probability of deepest penetration as a function of time. In the present theory, we take this penetration probability from the Pearson-Helfand theory and apply it to a randomly cross-linked polymer network. This is accomplished by performing a suitable averaging over the distribution of dangling-chain lengths.
Theoretical Background From numerous experiment~l-~ on the long-term stress relaxation and creep of elastomers, it has been found that an excellent representation of the data in the long-time limit is given by the Thirion-Chasset equation2
E ( t ) = E,[1
+ (t/rE)-"'],
( t / r E )>> 1
(la) X
for the tensile stress relaxation modulus E ( t ) and the analogous expression
D(t)= l/E(t),
t
-
m
x
Ob)
for the tensile creep compliance D ( t ) ,for m
